A Higher-Order Improved Runge–Kutta Method and Cubic B-Spline Approximation for the One-Dimensional Nonlinear RLW Equation

Abstract

This article developed a significant improvement of a Galerkin-type approximation to the regularized long-wave equation (RLW) solution under homogeneous Dirichlet boundary conditions for achieving higher accuracy in time variables. First, a basis derived from cubic B-splines and limit conditions is used to perform a Galerkin-type approximation. Then, a Crank–Nicolson and fourth-order 4-stage improved Runge–Kutta scheme (IRK4) is used to discretize time. Both a strong stability analysis of a fully discrete IRK4 scheme and the evaluation of Von Neumann stability of the proposed Crank–Nicolson technique are examined. We demonstrate the efficiency of our method with two test problems. The analytical and numerical solutions found in the literature are then contrasted with the approximate solutions produced by the suggested method. The validated numerical results illustrate that the provided technique is more efficient and converges faster than earlier research, resulting in less computational time, smaller space dimensions, and storage. As a result, the proposed numerical approach is appealing for approximating PDEs whose explicit solution is unknown for a variety of boundary conditions.